Fluid/Particle Interactions:

M5-Week 10: Flow Through Packed beds

• Porous medium

• Solid: stationary

• Liquid/gas: flow

• Resistance

Fluid/Particle Interactions:

Recall: Particle packing (shape, size, orientation,..)

•Voidage

•Sphericity

•Mean size

A measure of the ability of materials to transmit fluids

Permeability:

Fluid/Particle Interactions:

Packed Bed: Darcy’s law Darcy's Law (1856) defines the flow in a

porous medium: (water through sand)

v = κ ΔP / μ L

where:

v is the superficial fluid velocity through the medium

(Q/A)

μ is the viscosity of the fluid

ΔP is the applied pressure difference

L is the thickness of the medium

k is the permeability coefficient of a medium (unit:

area, m²)

L

Q

A

Fluid/Particle Interactions:

Kozeny-Carmen Equation

Kozeny 1927, 1933, Carman, 1937

key point is to calculate the pressure drop from fluid flow through the bed

Model packed bed as a bundle of pipes (capillaries)

Length of each pipe is KL: k=tortuosity)

For laminar flow start with Hagen-Poiseuille equation

or

Fluid/Particle Interactions:

Recall: Laminar flow in viscous pipe The Hagen-Poiseuille Equation

Laminar pipe flow equations:

pz

V

gH

pz

V

gH hp t l

1

1

1 11

2

2

2

2 22

2

2 2

2

2

21

1

1 zp

zp

hllhz

pz

p 2

2

21

1

1

2

32

D

lvP

L

DPQ

128

4

Relate l, v and d to pressure drop

Fluid/Particle Interactions:

Kozeny-Carmen Equation Consider the packed bed to be equivalent to many

tubes (equivalent diameters and lengths)

Consider voidage (e), shape of particles (y)

Result:

Pfr/L = 150(1-e) 2/e3 ●(Ug)/(ydv)2

Re = rUg(y dv)/ < 20)

Mono-sized particles: dv

For a size distribution, use specific surface mean

Fluid/Particle Interactions:

Sensitivity to parameters P proportional to U (laminar flow)

P inversely proportional to dv

2

How does P vary with voidage ?

Pfr/L=150(1-e) 2/ e3 ● (Ug)/(y dv)2

Fluid/Particle Interactions:

Turbulent Flow: Blake-Plummer Equation

For turbulent flow (high Reynolds numbers) the Blake-Plummer (empirical) equation is:

Pfr/L=1.75(1-e)/e3 ● rUg2/(ydv)

P proportional to Ug2

P inversely proportional to dv

Fluid/Particle Interactions:

Ergun Equation

Combine equations Kozeny-Carman and Blake-Plummer

to cover the whole range of Reynolds numbers: Ergun equation (1952)

Pfr/L=150(1-e)2/e3●(Ug )/(y dv)2 + 1.75(1-e)/e3● rUg

2/(ydv)

Viscous losses Turbulent losses

Re < 20, viscous loss term dominates

Re > 1000, Turbulent loss term dominates

Fluid/Particle Interactions:

Ergun Equation

• Use the Ergun Equation for all packed bed pressure drop

calculations

Friction factor: ff = 150 /Re +1.75

ff = P dv e3/r (1-e) Ug

2 L , Re = dvUg r/ (1-e)

Fluid/Particle Interactions:

Packed Bed: Darcy’s law

Kozeny-Carmen Equation

Blake-Plummer Equation

Ergun Equation

Fluid/Particle Interactions:

Example A packed bed of solid particles occupies a 1-m long

cylinder of cross-section 0.04 m2, The mass of particles in the bed is 50 kg, the surface-volume mean diameter of particles is 1 mm, A liquid flows upwards through the bed,

1. Calculate the voidage e of the bed,

2. When the volume rate of liquid is 1.44 m3/h, Calculate the pressure drop of the bed.

dv = 1 mm rP = 2500 kg/m3

= 0.002 Pa.s r = 850 kg/m3

y 1 A: 0.04 m2

Q = 1.44m3/h, L = 1m

Fluid/Particle Interactions:

Flow Through Packed beds: applications

Filtration

Gas/solid reactors

Catalytic reactors

Ion-exchange/adsorption

Flow through soils, waste dumps

Oil reservoir engineering,

Packed column/towers

Fluid/Particle Interactions:

Example: Filtration

Low velocity flow through,

particles be retained by the filter medium

A continuous deposition of solids

Resistance to flow increase through the operation

Pressure drop: size, voidage

Fluid/Particle Interactions:

Filtration

Incompressible cake

(From Ergun equation)

( Particles form cake with constant voidage)

Pfr/L= rc Ug

The volume ratio of cake formed by the passage of unit volume of filtrate:

* Here V is the volume of filtrate (liquid) passed in a time t

Instantaneous Volumetric flow rate dV/dt at time t:

Fluid/Particle Interactions:

• Including the resistance of the filter medium

Normally: Filter medium resistance rm is expressed as the equivalent thickness of the cake Heq

Veq is the volume of the filtrate that must pass in order to create a cake of thickness Heq

Note: here H is the length of the packed bed (cake), same as L

Time-volume relation:

Fluid/Particle Interactions:

Washing the cake

Removal of filtrate during washing of the filter cake

Fluid/Particle Interactions:

Compressible cake: (volidage changes, cake resistance increases)

Analysis of the pressure drop-flow relationship for a compressible cake

In practice: the pressure/resistance relation should be found in experiments

Appling Ergun Eq. laminar flow

Ps: pressure difference

Fluid/Particle Interactions:

Fluid/Particle Interactions:

Application: Ion-exchange Column

Ion-exchangeable species (resin, etc)

Liquid fed from the top, ion-exchange occurs

Purification/environmental purpose

Fluid/Particle Interactions:

Application: Packed column/towers

Shaped particles packed in a column

Used for bring two phases in contact with one-another (G-L, L-L systems, etc.)

Characteristics:

generally the size of packing elements in

columns large, Re large, turbulent

Packing elements have large internal surfaces, high flow resistance

Fluid/Particle Interactions:

Summary of Packed Beds

– Features of packed beds

– Ergun equation: relationship between pressure drop and flow rate

– Application examples